The survey, published yesterday by the Chartered Institute of Purchasing and Supply, also showed a significant deceleration of growth in new business for companies in the services sector.
CIPS's business activity index for the UK services sector dipped from 58.8 in December to 58.3 in January on a seasonally-adjusted basis, but remained well above the level of 50 which is deemed to separate expansion from contraction.
The City had projected a headline reading of 59 from the latest purchasing managers' index (PMI) survey for services.
Growth in services activity has now eased for three consecutive months, with the main index having dropped from 62.5 in October. CIPS's services survey does not include retailers.
The new business index for services dropped to 58.9 in January, from 62.1 in December. This signals the slowest pace of growth in new business in the services sector for eight months.
A survey published on Monday by CIPS showed that growth in the UK manufacturing sector eased further in January.
However, CIPS's latest survey of UK construction activity, published on Tuesday, showed that growth in this sector accelerated in January to its fastest monthly pace since August 2007. Housebuilding activity has surged on the back of moves by the Coalition Government to boost the UK residential property market.
Figures from the Office for National Statistics last month showed the pace of UK economic growth eased to 0.7% quarter-on-quarter in the final three months of 2013. Gross domestic product, according to the ONS, rose by 0.8% in each of the preceding two quarters.
In spite of this growth in recent quarters, economic output in the UK, in contrast to other major developed economies such as the US and Germany, remains below its peak ahead of the Great Recession of 2008/09.
Samuel Tombs, UK economist at consultancy Capital Economics, said of the further easing of services growth in CIPS's survey: "The further deterioration of the services survey in January, after the slight slowdown in GDP growth in Q4, may raise concerns that the recovery is starting to lose momentum. But there are many reasons to remain optimistic.
"We think it more likely that the decline in the PMI (purchasing managers' index) has simply moved it back towards more sensible levels."
In spite of the further easing of services growth, confidence among companies in this sector about activity levels over the coming 12 months climbed to the highest level since March 2010.
Chris Williamson, chief economist at survey compiler Markit, said: "The service sector saw another month of strong growth in January, suggesting that the UK economy continues to recover at a rapid pace. Although the pace of expansion slowed, we must remember that growth was exceptionally strong in previous months, and also that parts of the country saw record rainfall in January."
He added: "With business optimism about the future reaching the highest for almost four years, we should see growth revive again in February, hopefully as the weather improves and households and businesses dry out from January's rainfall.
"However, even with the easing seen in January, the sector is still expanding at a rate that bodes well for another strong GDP reading in the first quarter. Taken together, the three PMI surveys are signalling quarterly GDP growth of 0.8%."